Probability Conditional Probability and Independence

1
Probability

• Conditional Probability
• and
• Independence

2
Importance of the Sample Space

• The probability of an event depends on the sample
  space in question. The sample space is critical
  to determining the probability.
• In certain types of probabilistic situations, the
  entire sample space is not utilized only a
  portion is needed.

3
Rolling a Single Die

• Consider rolling a single die.
• List the sample space.
• What is the probability of rolling a 5?
• What is the probability of rolling a 5 given that
  an odd number has been rolled?

4
Conditional Probability

• Let E and F be events is a sample space S. The
  conditional probability Pr(E F) is the
  probability of event E occurring given the
  condition that event F has occurred. In
  calculating this probability, the sample space is
  restricted to F.

provided that Pr(F) ? 0.
5
Example Conditional Probability

• Twenty percent of the employees of Acme Steel
  Company are college graduates. Of all its
  employees, 25 earn more than 50,000 per year,
  and 15 are college graduates earning more than
  50,000. What is the probability that an employee
  selected at random earns more than 50,000 per
  year, given that he or she is a college graduate?

6
Conditional Probability Equally Likely Outcomes

• Conditional Probability in Case of Equally Likely
  Outcomes

provided that number of outcomes in F ? 0.
7
Example Conditional Probability

• A sample of two balls are selected from an urn
  containing 8 white balls and 2 green balls. What
  is the probability that the second ball selected
  is white given that the first ball selected was
  white?

8
Product Rule

• Product Rule If Pr(F) ? 0,
• Pr(E n F) Pr(F) ? Pr(E F).
• The product rule can be extended to three events.
• Pr(E1 n E2 n E3) Pr(E1) ? Pr(E2 E1) ? Pr(E3
  E1 n E2)

9
Example Product Rule

• A sequence of two playing cards is drawn at
  random (without replacement) from a standard deck
  of 52 cards. What is the probability that the
  first card is red and the second is black?
• Let
• F "the first card is red," and
• E "the second card is black."

10
Independent Events

• Events are said to be independent if the
  probability of one event does not affect the
  likelihood of occurrence of the other event(s).
• A collection of events is said to be independent
  if for each collection of events chosen from
  them, the probability that all the events occur
  equals the product of the probabilities that each
  occurs.

11
Independence

• Let E and F be events. We say that E and F are
  independent provided that
• Pr(E n F) Pr(E) ? Pr(F).
• Equivalently, they are independent provided that
• Pr(E F) Pr(E) and Pr(F E) Pr(F).

12
Example Independence

• Let an experiment consist of observing the
  results of drawing two consecutive cards from a
  52-card deck.
• Let E "second card is black" and
• F "first card is red".
• Are these two events independent?

13
Independence of a Set of Events

• A set of events is said to be independent if, for
  each collection of events chosen from them, say
  E1, E2, , En, we have
• Pr(E1 n E2 n n En) Pr(E1) ? Pr(E2) ??
  Pr(En).

14
Example Independence of a Set

• A company manufactures stereo components.
  Experience shows that defects in manufacture are
  independent of one another. Quality control
  studies reveal that
• 2 of CD players are defective, 3 of amplifiers
  are defective, and 7 of speakers are defective.
• A system consists of a CD player, an amplifier,
  and 2 speakers. What is the probability that the
  system is not defective?

15
Example 1

• The proportion of individuals in a certain city
  earning more than 35,000 per year is .25. The
  proportion of individuals earning more than
  35,000 and having a college degree is .10.
  Suppose that a person is randomly chosen and he
  turns out to be earning more than 35,000. What
  is the probability that he is a college graduate?

16
Example 2

• A stereo system contains 50 transistors. The
  probability that a given transistor will fail in
  100,000 hours of use is .0005. Assume that the
  failures of the various transistors are
  independent of one another. What is the
  probability that no transistor will fail during
  the first 100,000 hours of use?

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Example 3

• Let E and F be events with P(E) .3, P(F) .6,
  and P(E ? F) .7 . Find
• a.) P(E n F)
• b.) P(E F)
• c.) P(F E)
• d.) P(E? n F)
• e.) P(E ? F)

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Example 4

• Of the students at a certain college, 50
  regularly attend the football games, 30 are
  first-year students, and 40 are upper-class
  students who do not regularly attend football
  games. Suppose that a student is selected at
  random.
• a.) What is the probability that the person both
  is a first-year student and regularly attends
  football games?

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Example 4 (continued)

• Of the students at a certain college, 50
  regularly attend the football games, 30 are
  first-year students, and 40 are upper-class
  students who do not regularly attend football
  games. Suppose that a student is selected at
  random.
• b.) What is the probability that the person
  regularly attends football games given that he is
  a first-year student?
• c.) What is the probability that the person is a
  first-year student given that he regularly
  attends football games?

20
Example 5

• Two poker chips are selected at random from an
  bag containing two white chips and three red
  chips. What is the probability that both
  chips are white given that at least one of them
  is white?

21
Example 6

• Out of 250 students interviewed at a community
  college, 90 were taking mathematics but not
  computer science, 160 were taking mathematics,
  and 50 were taking neither mathematics nor
  computer science. Find the probability that a
  student chosen at random was
• a.) taking just computer science.

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Example 6

• Out of 250 students interviewed at a community
  college, 90 were taking mathematics but not
  computer science, 160 were taking mathematics,
  and 50 were taking neither mathematics nor
  computer science. Find the probability that a
  student chosen at random was
• b.) taking mathematics or computer science, but
  not both.
• c.) taking mathematics, given that the student
  was taking computer science.
• d.) taking computer science, given that the
  student was not taking mathematics.

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Example 7

• Find the probability that a student selected at
  random is
• a.) a senior.
• b.) working full-time.
• c.) working part-time, given
• that the student is a first-
• year student.
• d.) a junior or senior, given
• that the student does not
• work.

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Example 8

• A bag contains eight purple marbles, six blue
  marbles, and 12 red marbles. Four marbles are
  selected from the bag.
• a.) What is the probability that they are all
  purple?
• b.) What is the probability that they are all
  the same color?